MCAT Physics and Math Review

Chapter 4: Fluids

Introduction

Hidden beneath the waves of the Mediterranean Sea, at depths of more than 4,000 meters, lie three lakes. The water in these “seas under the sea” is so salty—five to ten times saltier than the seawater that sits above it—that its extreme density prevents it from mixing with the ocean water above, forming a layer of separation not unlike that between the oil and water in a bottle of salad dressing. These underwater lakes behave eerily like their more common cousins found at sea level. They have tides, shore lines, beach ridges, and swash zones. When deep sea exploratory vessels set down on their surfaces, the vessels bob up and down, causing ripples to emanate outward like a stone dropped in a pond.

Suboceanic lakes and rivers present a particularly fascinating opportunity to illustrate the physics of fluids and solids. This chapter covers the important concepts and principles of fluid mechanics as they are tested on the MCAT. We will begin with a review of some important terms and measurements, including density and pressure. Our next topic will be hydrostatics, the branch of fluid mechanics that characterizes the behavior of fluids at rest. We’ll then turn our attention to fluid dynamics, including Bernoulli’s equation and the aerodynamics of flight. Finally, the chapter concludes with a discussion of fluid dynamics in physiology, examining the properties that motivate the movement of blood and air within the body.

4.1 Characteristics of Fluids and Solids

Fluids are characterized by their ability to flow and conform to the shapes of their containers. Solids, on the other hand, do not flow and are rigid enough to retain a shape independent of their containers. Both liquids and gases are fluids. The natural gas (methane) that many of us use to cook flows through pipes to the burners of our stove and ovens, and the air that we breathe flows in and out of our lungs, filling the spaces of our respiratory tract and the alveoli.

Fluids and solids share certain characteristics. Both can exert forces perpendicular to their surface, although only solids can withstand shear (tangentialforces. Fluids can impose large perpendicular forces; falling into water from a significant height can be just as painful as falling onto a solid surface.

DENSITY

All fluids and solids are characterized by the ratio of their mass to their volume. This is known as density, which is a scalar quantity and therefore has no direction. The equation for density is

Equation 4.1

where ρ (rho) represents density, m is mass, and V is volume. The SI units for density are  but you may find it convenient to use  or  both of which may be seen on the MCAT. Remember that a milliliter and a cubic centimeter are the same volume. A word of caution: students sometimes assume that if the mL and the cm3 are equivalent, then so must be the liter and the m3. This is absolutely not the case; in fact, there are 1000 liters in a cubic meter. For the MCAT, it is important to know the density of water, which is 

The weight of any volume of a given substance with a known density can be calculated by multiplying the substance’s density by its volume and the acceleration due to gravity. This is a calculation that appears frequently when working through buoyancy problems on Test Day:

Fg = ρVg

Equation 4.2

The density of a fluid is often compared to that of pure water at 1 atm and 4°C, a variable called specific gravity. It is at this combination of pressure and temperature that water has a density of exactly  The specific gravity is given by

Equation 4.3

This is a unitless number that is usually expressed as a decimal. The specific gravity can be used as a tool for determining if an object will sink or float in water, as described later in this chapter.

MCAT EXPERTISE

If an object’s density is given in  , its specific gravity is simply its density as a dimensionless number. This is because the density of water in  is 1.

Example:

Find the specific gravity of benzene, given that its density is 

Solution:

The ratio of the density of benzene to the density of water is the specific gravity. Either the numerator must be converted to  or the denominator (the density of water) must be given in 

PRESSURE

Pressure is a ratio of the force per unit area. The equation for pressure is

Equation 4.4

where P is pressure, F is the magnitude of the normal force vector, and A is the area. The SI unit of pressure is the pascal (Pa), which is equivalent to the newton per square meter  Other commonly used units of pressure are millimeters of mercury (mmHg), torr, and the atmosphere (atm). Millimeters of mercury and torr are identical units. The unit of atmosphere is based on the average atmospheric pressure at sea level. The conversions between Pa, mmHg, torr, and atm are as follows:

1.013 × 105 Pa = 760 mmHg ≡ 760 torr = 1 atm

MCAT EXPERTISE

If you ever forget the units of a variable, you can derive them from equations. You know that pressure equals force over area. Because you know the units of force (N) and area (m2), you can solve for the base units of pascal by plugging these units into the equation: 

Pressure is a scalar quantity, and therefore has a magnitude but no direction. It is easy to assume that pressure has a direction because it is related to a force, which is a vector. However, note that it is the magnitude of the normal force that is used. No matter where one positions a given surface, the pressure exerted on that surface within a closed container will be the same, neglecting gravity. For example, if we placed a surface inside a closed container filled with gas, the individual molecules, which are moving randomly within the space, will exert pressure that is the same at all points within the container. Because the pressure is the same at all points along the walls of the container and within the space of the container itself, pressure applies in all directions at any point and, therefore, is a scalar rather than a vector. Of course, because pressure is a ratio of force to area, when unequal pressures are exerted against objects, the forces acting on the object will add in vectors, possibly resulting in acceleration. It’s this difference in pressure that causes air to rush into and out of the lungs during respiration, windows to burst outward during a tornado, and the plastic covering a broken car window to bubble outward when the car is moving. Note that when gravity is present, this also results in a pressure differential, which we will explore with hydrostatics later in this chapter.

Example:

The window of a skyscraper measures 2.0 m by 3.5 m. If a storm passes by and lowers the pressure outside the window to 0.997 atm while the pressure inside the building remains at 1 atm, what is the net force pushing on the window?

Solution:

Because the pressures are different on the two sides of this window, there will be a net force pushing on it in the direction of the lower pressure (outside the window). The difference in pressure itself can be used to determine the net force:

Absolute Pressure

At this very moment, countless trillions of air molecules are exerting tremendous pressure on our bodies, with a total force of about 2 × 105 N! Of course, we don’t actually feel all this pressure because our internal organs exert a pressure that perfectly balances it.

Atmospheric pressure changes with altitude. Residents of Denver (5280 feet above sea level) experience atmospheric pressure equal to 632 mmHg (0.83 atm), whereas travelers making their way through Death Valley (282 feet below sea level) experience atmospheric pressure equal to 767 mm Hg (1.01 atm). Atmospheric pressure impacts a number of processes, including hemoglobin’s affinity for oxygen and the boiling of liquids.

Absolute (hydrostaticpressure is the total pressure that is exerted on an object that is submerged in a fluid. Remember that fluids include both liquids and gases. The equation for absolute pressure is

P = Po + ρgz

Equation 4.5

where P is the absolute pressure, Po is the incident or ambient pressure (the pressure at the surface), ρ is the density of the fluid, g is acceleration due to gravity, and z is the depth of the object. Do not make the mistake of assuming that Po always stands for atmospheric pressure. In open air and most day-to-day situations Po is equal to 1 atm, but in other fluid systems, the surface pressure may be higher or lower than atmospheric pressure. In a closed container, such as a pressure cooker, the pressure at the surface may be much higher than atmospheric pressure. This is, in fact, exactly the point of a pressure cooker, which allows food to cook at higher temperatures. This is because the increased pressure raises the boiling point of water in the food, thus reducing the cooking time and preventing loss of moisture.

REAL WORLD

A useful way to remember the two parts of the absolute pressure equation is to think of diving into a swimming pool. At the surface of the water, the absolute pressure is equal to the atmospheric pressure (Po). But if you dive into the pool, the water exerts an extra pressure on you (ρgz), in addition to the surface pressure. You feel this extra pressure on your eardrums.

Gauge Pressure

When you check the pressure in your car or bike tires using a device known as a gauge, you are measuring the gauge pressure, which is the difference between the absolute pressure inside the tire and the atmospheric pressure outside the tire. In other words, gauge pressure is the amount of pressure in a closed space above and beyond atmospheric pressure. This is a more common pressure measurement than absolute pressure, and the equation is:

Pgauge = P – Patm = (Po + ρgz) – Patm

Equation 4.6

Note that when Po = Patm, then Pgauge = P – Po = ρgz at a depth z.

Example:

A diver in the ocean is 20 m below the surface. What is the absolute pressure she experiences? What is the gauge pressure at her depth? (Note: The density of sea water is  )

Solution:

First, we use the equation for absolute pressure in a liquid:

Now, use the equation for gauge pressure:

Pgauge = P − Patm = (3.02 − 1.013) × 105 Pa = 2.01 × 105 Pa

MCAT Concept Check 4.1:

Before you move on, assess your understanding of the material with these questions.

1.    How does gauge pressure relate to the pressure exerted by a column of fluid?

2.    What is the relationship between weight and density?

3.    What is the SI unit for pressure? What are other common units of pressure?

·        SI unit:

·        Other units:

4.    True or False: Density is a scalar quantity.